The quest to explore and utilize marine resources has propelled the development of sophisticated underwater robotic systems. Among these, the Underwater Vehicle-Manipulator System (UVMS) represents a critical technological integration, combining the mobility of an autonomous underwater vehicle (AUV) with the dexterity of a robotic manipulator. The core of many underwater intervention tasks—such as scientific sampling, infrastructure inspection, or object recovery—hinges on the precise control of the manipulator’s end effector. Accurate trajectory tracking of this end effector in the presence of complex hydrodynamic forces, model uncertainties, and external ocean disturbances remains a formidable challenge. This article addresses this challenge by presenting a novel, robust control strategy designed to guarantee precise end effector tracking within a fixed, predictable time frame, independent of the system’s initial conditions.
The dynamic environment of the ocean subjects UVMS to unmodeled dynamics, parameter variations, and time-varying disturbances like currents. These factors can severely degrade the performance of conventional control methods, leading to tracking errors that compromise task success. While control techniques like Proportional-Integral-Derivative (PID), adaptive control, and neural networks have been applied, each has limitations concerning robustness, convergence speed, or dependency on extensive tuning. Sliding Mode Control (SMC) is renowned for its robustness against matched uncertainties and disturbances. However, combining SMC with a disturbance observer can significantly enhance performance by actively estimating and compensating for lumped disturbances.
Recently, the concepts of finite-time and fixed-time stability have gained prominence in control theory. While finite-time control ensures convergence within a finite duration that depends on initial conditions, fixed-time control guarantees convergence within a bounded time that is independent of the initial state. This property is highly desirable for robotic systems operating in unpredictable environments, as it provides a guaranteed upper bound on the settling time. This work leverages this property to develop a Fixed-Time Disturbance Observer (FTDO) that can accurately estimate the lumped disturbances affecting the UVMS in a fixed time. This estimate is then fed into an Integral Sliding Mode Controller (ISMC), which is also designed with fixed-time convergence characteristics for the sliding surface. The synthesis, termed FTDO-ISMC, ensures that the UVMS end effector trajectory tracking error converges to zero within a fixed time, offering superior robustness and predictable performance.
Mathematical Preliminaries and System Modeling
Key Definitions and Lemmas
We first recall essential definitions and lemmas related to fixed-time stability.
Definition 1 (Finite-Time Stability): Consider the nonlinear system:
$$\dot{x} = f(x(t)), \quad x(0)=x_0,$$
where $x \in \mathbb{R}^n$ is the state vector. The system is finite-time stable if it is asymptotically stable and there exists a function $T(x_0): \mathbb{R}^n \rightarrow \mathbb{R}_{>0}$, called the settling-time function, such that for every $x_0$, the solution $x(t)$ reaches the origin at time $T(x_0)$, i.e., $x(t)=0$ for all $t \ge T(x_0)$.
Definition 2 (Fixed-Time Stability): The system above is fixed-time stable if it is finite-time stable and the settling-time function $T(x_0)$ is bounded by some positive constant $T_{\text{max}}$, i.e., $T(x_0) \le T_{\text{max}}$ for all $x_0 \in \mathbb{R}^n$.
Lemma 1: For the scalar system:
$$\dot{y} = -k_1 y^{m_1} – k_2 y^{m_2}, \quad y(0)=y_0,$$
where $k_1, k_2 > 0$, $m_1 = p/q$, $m_2 = r/q$, with $p, q, r$ being positive odd integers satisfying $p < q < r$, the origin is fixed-time stable. The convergence time $T$ is bounded by:
$$T \le T_{\max} = \frac{1}{k_1} \frac{q}{q-p} + \frac{1}{k_2} \frac{q}{r-q}.$$
Lemma 2: Consider the double-integrator system $\dot{x}_1 = x_2$, $\dot{x}_2 = u$. The control law:
$$u = -l_1 \text{sig}^{\alpha_1}(x_1) – l_2 \text{sig}^{\alpha_2}(x_2),$$
with $l_1, l_2 > 0$, $\alpha_1 \in (0,1)$, $\alpha_2 = 2\alpha_1/(1+\alpha_1)$, and $\text{sig}^{\alpha}(z)=|z|^\alpha \text{sign}(z)$, ensures finite-time stability of the origin.
UVMS Kinematic and Dynamic Model
The UVMS is modeled as a rigid body (the vehicle) with a serial-link manipulator attached. Two coordinate frames are defined: the inertial frame $\{E\}$ and the body-fixed frame $\{B\}$ attached to the vehicle’s center of mass. The system’s configuration is described by the generalized coordinate vector $\boldsymbol{\eta} = [\boldsymbol{\eta}_1^T, \boldsymbol{\eta}_2^T, \boldsymbol{\eta}_3^T]^T$, where $\boldsymbol{\eta}_1 = [x, y, z]^T \in \mathbb{R}^3$ is the end effector position in $\{E\}$, $\boldsymbol{\eta}_2 = [\phi, \theta, \psi]^T \in \mathbb{R}^3$ are the vehicle’s Euler angles (roll, pitch, yaw), and $\boldsymbol{\eta}_3 = [\theta_1, \theta_2]^T \in \mathbb{R}^2$ are the manipulator’s joint angles. For control design, a 4-degree-of-freedom (DOF) vehicle model (surge, sway, heave, yaw) coupled with a 2-DOF manipulator is considered, giving $\boldsymbol{\eta} \in \mathbb{R}^6$.
The generalized velocity vector is $\boldsymbol{\upsilon} = [\boldsymbol{\upsilon}_1^T, \boldsymbol{\upsilon}_2^T]^T$, where $\boldsymbol{\upsilon}_1 = [u, v, w, r]^T \in \mathbb{R}^4$ contains the vehicle’s linear and angular velocities in $\{B\}$, and $\boldsymbol{\upsilon}_2 = [\dot{\theta}_1, \dot{\theta}_2]^T \in \mathbb{R}^2$ are the joint velocities. The kinematic relationship is:
$$\dot{\boldsymbol{\eta}} = \boldsymbol{J}(\boldsymbol{\eta}) \boldsymbol{\upsilon},$$
where $\boldsymbol{J}(\boldsymbol{\eta}) \in \mathbb{R}^{6 \times 6}$ is the geometric Jacobian matrix mapping body-fixed velocities to inertial-frame end effector and orientation rates.
The dynamic model in the body-fixed frame is given by:
$$\boldsymbol{M} \dot{\boldsymbol{\upsilon}} + \boldsymbol{C}(\boldsymbol{\upsilon})\boldsymbol{\upsilon} + \boldsymbol{D}(\boldsymbol{\upsilon})\boldsymbol{\upsilon} + \boldsymbol{g}(\boldsymbol{\eta}) = \boldsymbol{\tau} + \boldsymbol{\tau}_{\text{ext}},$$
where:
- $\boldsymbol{M} \in \mathbb{R}^{6 \times 6}$ is the inertia matrix (including added mass), symmetric and positive definite.
- $\boldsymbol{C}(\boldsymbol{\upsilon}) \in \mathbb{R}^{6 \times 6}$ is the Coriolis and centripetal matrix.
- $\boldsymbol{D}(\boldsymbol{\upsilon}) \in \mathbb{R}^{6 \times 6}$ is the damping matrix.
- $\boldsymbol{g}(\boldsymbol{\eta}) \in \mathbb{R}^{6}$ is the vector of gravitational/buoyancy forces and moments.
- $\boldsymbol{\tau} \in \mathbb{R}^{6}$ is the vector of control inputs (thruster forces and joint torques).
- $\boldsymbol{\tau}_{\text{ext}} \in \mathbb{R}^{6}$ represents the lumped effect of unmodeled dynamics and external disturbances.
For controller design, it is practical to express the dynamics in the inertial (task) space where the end effector trajectory is defined. Differentiating the kinematics and substituting the dynamics yields:
$$\ddot{\boldsymbol{\eta}} = \boldsymbol{J} \dot{\boldsymbol{\upsilon}} + \dot{\boldsymbol{J}} \boldsymbol{\upsilon} = \boldsymbol{F}(\boldsymbol{\eta}, \dot{\boldsymbol{\eta}}) + \boldsymbol{G}(\boldsymbol{\eta}) \boldsymbol{\tau} + \boldsymbol{d},$$
where $\boldsymbol{F}(\boldsymbol{\eta}, \dot{\boldsymbol{\eta}})$ encapsulates the known dynamics (Coriolis, damping, gravity), $\boldsymbol{G}(\boldsymbol{\eta}) = \boldsymbol{J} \boldsymbol{M}^{-1}$, and $\boldsymbol{d} = \boldsymbol{J} \boldsymbol{M}^{-1} \boldsymbol{\tau}_{\text{ext}}$ represents the lumped disturbance in task space. We assume this disturbance is bounded, i.e., $\|\boldsymbol{d}\| \le L_d$.
Controller Design: FTDO-ISMC Synthesis
The control objective is to force the actual end effector trajectory $\boldsymbol{\eta}(t)$ to track a desired, twice-differentiable trajectory $\boldsymbol{\eta}_d(t)$ within a fixed time, despite the presence of the lumped disturbance $\boldsymbol{d}$. Define the tracking errors:
$$\boldsymbol{e}_1 = \boldsymbol{\eta} – \boldsymbol{\eta}_d, \quad \boldsymbol{e}_2 = \dot{\boldsymbol{\eta}} – \dot{\boldsymbol{\eta}}_d.$$
The error dynamics are:
$$\dot{\boldsymbol{e}}_1 = \boldsymbol{e}_2,$$
$$\dot{\boldsymbol{e}}_2 = \boldsymbol{F}(\boldsymbol{\eta}, \dot{\boldsymbol{\eta}}) + \boldsymbol{G}(\boldsymbol{\eta}) \boldsymbol{\tau} + \boldsymbol{d} – \ddot{\boldsymbol{\eta}}_d.$$
Step 1: Fixed-Time Disturbance Observer (FTDO) Design
To actively compensate for $\boldsymbol{d}$, we design a FTDO. Let $\boldsymbol{z}_1$ and $\boldsymbol{z}_2$ be estimates of $\boldsymbol{e}_2$ and $\boldsymbol{d}$, respectively. Define observation errors $\boldsymbol{\varepsilon}_1 = \boldsymbol{z}_1 – \boldsymbol{e}_2$ and $\boldsymbol{\varepsilon}_2 = \boldsymbol{z}_2 – \boldsymbol{d}$. The FTDO is proposed as:
$$\dot{\boldsymbol{z}}_1 = -\boldsymbol{\kappa}_1 \text{sig}^{\gamma_1}(\boldsymbol{\varepsilon}_1) – \boldsymbol{\kappa}_2 \text{sig}^{\gamma_2}(\boldsymbol{\varepsilon}_1) + \boldsymbol{z}_2 + \boldsymbol{\phi},$$
$$\dot{\boldsymbol{z}}_2 = -\boldsymbol{\kappa}_3 \text{sig}^{2\gamma_1-1}(\boldsymbol{\varepsilon}_1) – \boldsymbol{\kappa}_4 \text{sig}^{2\gamma_2-1}(\boldsymbol{\varepsilon}_1),$$
where $\boldsymbol{\phi} = \boldsymbol{F}(\boldsymbol{\eta}, \dot{\boldsymbol{\eta}}) + \boldsymbol{G}(\boldsymbol{\eta}) \boldsymbol{\tau} – \ddot{\boldsymbol{\eta}}_d$, $\boldsymbol{\kappa}_i$ ($i=1,2,3,4$) are positive definite diagonal gain matrices, $0 < \gamma_2 < 1$, $\gamma_1 > 1$, and $\text{sig}^\alpha(\boldsymbol{x})$ is applied element-wise. According to Lemma 1 and related fixed-time stability results, the observer errors $(\boldsymbol{\varepsilon}_1, \boldsymbol{\varepsilon}_2)$ converge to zero in a fixed time $T_o$ bounded by a constant independent of initial observation errors. Thus, after $T_o$, we have an accurate estimate: $\boldsymbol{z}_2 = \boldsymbol{d}$.
Step 2: Integral Sliding Mode Control with Fixed-Time Convergence
We now design an ISMC law that utilizes the disturbance estimate. First, define an integral sliding surface $\boldsymbol{s} \in \mathbb{R}^6$:
$$\boldsymbol{s} = \boldsymbol{e}_2 + \boldsymbol{\beta}_1 \int_0^t \text{sig}^{\alpha_1}(\boldsymbol{e}_1(\tau)) d\tau + \boldsymbol{\beta}_2 \int_0^t \text{sig}^{\alpha_2}(\boldsymbol{e}_2(\tau)) d\tau,$$
where $\boldsymbol{\beta}_1, \boldsymbol{\beta}_2 > 0$ are diagonal gain matrices, and $\alpha_1, \alpha_2$ are chosen as in Lemma 2 (e.g., $\alpha_1=1/2, \alpha_2=2/3$). This surface incorporates both error states and ensures finite-time convergence of $\boldsymbol{e}_1, \boldsymbol{e}_2$ to zero once on the surface $\boldsymbol{s}=0$.
Differentiating $\boldsymbol{s}$ and substituting the error dynamics yields:
$$\dot{\boldsymbol{s}} = \dot{\boldsymbol{e}}_2 + \boldsymbol{\beta}_1 \text{sig}^{\alpha_1}(\boldsymbol{e}_1) + \boldsymbol{\beta}_2 \text{sig}^{\alpha_2}(\boldsymbol{e}_2) = \boldsymbol{F} + \boldsymbol{G}\boldsymbol{\tau} + \boldsymbol{d} – \ddot{\boldsymbol{\eta}}_d + \boldsymbol{\beta}_1 \text{sig}^{\alpha_1}(\boldsymbol{e}_1) + \boldsymbol{\beta}_2 \text{sig}^{\alpha_2}(\boldsymbol{e}_2).$$
The control law is designed as:
$$\boldsymbol{\tau} = \boldsymbol{G}^{-1}(\boldsymbol{\eta}) \left[ -\boldsymbol{F} + \ddot{\boldsymbol{\eta}}_d – \boldsymbol{\beta}_1 \text{sig}^{\alpha_1}(\boldsymbol{e}_1) – \boldsymbol{\beta}_2 \text{sig}^{\alpha_2}(\boldsymbol{e}_2) – \boldsymbol{z}_2 – \boldsymbol{k}_1 \boldsymbol{s} – \boldsymbol{k}_2 \text{sig}^{\mu_1}(\boldsymbol{s}) – \boldsymbol{k}_3 \text{sig}^{\mu_2}(\boldsymbol{s}) \right],$$
where $\boldsymbol{k}_1, \boldsymbol{k}_2, \boldsymbol{k}_3 > 0$ are diagonal gain matrices, $0 < \mu_1 < 1$, and $\mu_2 > 1$. The term $-\boldsymbol{z}_2$ provides direct disturbance compensation based on the FTDO estimate. The remaining terms, particularly $- \boldsymbol{k}_2 \text{sig}^{\mu_1}(\boldsymbol{s}) – \boldsymbol{k}_3 \text{sig}^{\mu_2}(\boldsymbol{s})$, are the key to enforcing fixed-time convergence of the sliding variable $\boldsymbol{s}$ to zero.
Stability and Convergence Time Analysis
Theorem: Consider the UVMS error dynamics under the lumped disturbance $\boldsymbol{d}$ satisfying $\|\boldsymbol{d}\| \le L_d$. With the FTDO and the control law defined above, the closed-loop system is fixed-time stable. The end effector tracking errors $\boldsymbol{e}_1$ and $\boldsymbol{e}_2$ converge to zero in a fixed time $T_{\text{total}}$, whose upper bound is independent of the initial system state.
Proof Sketch: The proof proceeds in two phases using Lyapunov analysis.
Phase 1 (Fixed-Time Disturbance Estimation): Consider a Lyapunov function candidate for the observer error system, $V_o = \frac{1}{2} \boldsymbol{\varepsilon}_1^T \boldsymbol{\varepsilon}_1$. Its derivative, after substituting the observer dynamics and employing Lemma 1, can be shown to satisfy an inequality of the form:
$$\dot{V}_o \le -c_1 V_o^{\frac{\gamma_1+1}{2}} – c_2 V_o^{\frac{\gamma_2+1}{2}},$$
which guarantees that $V_o$ (and hence $\boldsymbol{\varepsilon}_1$, $\boldsymbol{\varepsilon}_2$) reaches zero in a fixed time $T_o \le T_{o,\max}$.
Phase 2 (Fixed-Time Convergence to Sliding Surface and Tracking): After $t \ge T_o$, we have $\boldsymbol{z}_2 = \boldsymbol{d}$. Now consider a Lyapunov function for the controlled system, $V_c = \frac{1}{2} \boldsymbol{s}^T \boldsymbol{s}$. Its derivative along the closed-loop trajectories becomes:
$$\dot{V}_c = \boldsymbol{s}^T \dot{\boldsymbol{s}} = \boldsymbol{s}^T \left( -\boldsymbol{k}_1 \boldsymbol{s} – \boldsymbol{k}_2 \text{sig}^{\mu_1}(\boldsymbol{s}) – \boldsymbol{k}_3 \text{sig}^{\mu_2}(\boldsymbol{s}) \right).$$
This simplifies to:
$$\dot{V}_c \le -k_{1,\min} \|\boldsymbol{s}\|^2 – k_{2,\min} \|\boldsymbol{s}\|^{\mu_1+1} – k_{3,\min} \|\boldsymbol{s}\|^{\mu_2+1},$$
where $k_{i,\min}$ are the smallest diagonal entries of $\boldsymbol{k}_i$. Noting that $\|\boldsymbol{s}\| = \sqrt{2 V_c}$, this inequality takes the form:
$$\dot{V}_c \le -\lambda_1 V_c – \lambda_2 V_c^{\frac{\mu_1+1}{2}} – \lambda_3 V_c^{\frac{\mu_2+1}{2}},$$
for positive constants $\lambda_1, \lambda_2, \lambda_3$. By Lemma 1, this ensures that the sliding surface $\boldsymbol{s} = 0$ is reached in a fixed time $T_s \le T_{s,\max}$, independent of initial errors.
Phase 3 (Finite-Time Convergence on Sliding Surface): Once on $\boldsymbol{s}=0$, the dynamics reduce to:
$$\boldsymbol{e}_2 = -\boldsymbol{\beta}_1 \text{sig}^{\alpha_1}(\boldsymbol{e}_1) – \boldsymbol{\beta}_2 \text{sig}^{\alpha_2}(\boldsymbol{e}_2).$$
This is precisely the form covered by Lemma 2, guaranteeing that $\boldsymbol{e}_1$ and $\boldsymbol{e}_2$ converge to zero in finite time $T_r$. The total convergence time is bounded by $T_{\text{total}} \le T_{o,\max} + T_{s,\max} + T_r$, which is a fixed constant. Thus, the end effector trajectory is tracked with precision within this guaranteed time frame. $\blacksquare$
Simulation Studies and Performance Evaluation
To validate the proposed FTDO-ISMC strategy, numerical simulations were conducted for a UVMS model comprising a 4-DOF vehicle and a 2-DOF manipulator. The model parameters used in the simulation are summarized in Table 1.
| Parameter | Value | Description |
|---|---|---|
| $m_v$ | 50 kg | Vehicle mass |
| $m_1, m_2$ | 5 kg, 3 kg | Manipulator link masses |
| $I_z, I_1, I_2$ | 40, 1.5, 0.8 kg·m² | Moments of inertia |
| $X_{\dot{u}}, Y_{\dot{v}}, …$ | 20, 30, 35, 15, 0.5, 0.3 kg | Added mass coefficients |
| $X_u, Y_v, Z_w, N_r$ | 25, 30, 30, 15 kg/s | Linear damping coefficients |
| $X_{u|u|}, Y_{v|v|}, …$ | 12, 15, 15, 8 kg/m | Quadratic damping coefficients |
| Link lengths | 0.8 m, 0.6 m | Manipulator arm lengths |
The controller parameters for the FTDO and the ISMC were tuned as follows:
| Component | Parameter | Value |
|---|---|---|
| FTDO | $\boldsymbol{\kappa}_1, \boldsymbol{\kappa}_2$ | $\text{diag}(5,5,5,5,5,5)$ |
| $\boldsymbol{\kappa}_3, \boldsymbol{\kappa}_4$ | $\text{diag}(10,10,10,10,10,10)$ | |
| $\gamma_1$ | 1.5 | |
| $\gamma_2$ | 0.5 | |
| ISMC | $\boldsymbol{\beta}_1, \boldsymbol{\beta}_2$ | $\text{diag}(2,2,2,2,2,2)$ |
| $\alpha_1, \alpha_2$ | 0.5, 2/3 | |
| $\boldsymbol{k}_1$ | $\text{diag}(8,8,8,8,8,8)$ | |
| $\boldsymbol{k}_2, \boldsymbol{k}_3$ | $\text{diag}(3,3,3,3,3,3)$ | |
| $\mu_1$ | 0.8 | |
| $\mu_2$ | 1.2 |
The desired trajectory for the end effector was a helical path combined with a changing orientation, defined by:
$$x_d(t) = 0.5t, \quad y_d(t) = 2\sin(0.2t), \quad z_d(t) = -1 + 0.5\cos(0.2t), \quad \psi_d(t)=0.1t, \quad \theta_{1d}(t)=0.2\sin(0.3t), \quad \theta_{2d}(t)=0.1\cos(0.3t).$$
A significant lumped disturbance $\boldsymbol{d}(t) = [2\sin(0.5t), 1.5\cos(0.3t), \sin(0.4t), 0.5\sin(0.6t), 0.8\cos(0.2t), 0.6\sin(0.25t)]^T$ was applied to test robustness. Three different initial states (both position and velocity) were simulated to demonstrate the fixed-time property.
The simulation results confirmed the theoretical predictions. For all three initial conditions, the end effector trajectory accurately converged to and followed the desired helical path. The tracking errors for both position ($\boldsymbol{e}_1$) and velocity ($\boldsymbol{e}_2$) converged to a small neighborhood of zero within approximately 4.5 seconds. Crucially, this convergence time was nearly identical for all different initial states, visually demonstrating the fixed-time convergence characteristic. The FTDO successfully estimated the complex, time-varying disturbance within the first second, enabling the controller to effectively cancel its effect.
To quantitatively highlight the superiority of the proposed FTDO-ISMC, its performance was compared against two other robust control strategies: a standard Non-singular Terminal SMC (NTSMC) and a Backstepping SMC (BSMC). The comparison was based on Integral Absolute Error (IAE) and Integral Time Absolute Error (ITAE) metrics for the end effector position tracking error over a 20-second simulation. The results are summarized in Table 3.
| Control Method | IAE (x,y,z) [m·s] | ITAE (x,y,z) [m·s²] | Key Observation |
|---|---|---|---|
| FTDO-ISMC (Proposed) | 0.142, 0.118, 0.095 | 0.851, 0.712, 0.573 | Fastest, most accurate convergence; fixed-time property. |
| NTSMC | 0.381, 0.402, 0.355 | 3.112, 3.245, 2.987 | Good robustness, but slower convergence and higher steady-state error. |
| BSMC | 0.523, 0.487, 0.511 | 5.234, 4.876, 5.102 | Largest error and slowest response due to lack of explicit disturbance estimation. |
The data clearly shows that the proposed FTDO-ISMC achieves significantly lower IAE and ITAE values, indicating not only smaller tracking error magnitude but also much faster error reduction. This quantifies the advantage gained from the fixed-time disturbance observer and the fixed-time convergent sliding mode control law.
Conclusion
This article has presented a comprehensive solution for the precise trajectory tracking control of an underwater vehicle-manipulator system’s end effector. The proposed FTDO-ISMC strategy effectively addresses the dual challenges of model uncertainty/external disturbances and the need for predictable, fast convergence. The fixed-time disturbance observer provides accurate estimation of lumped disturbances within a bounded time independent of initial observation errors. This estimate is seamlessly integrated into a fixed-time convergent integral sliding mode control law, which guarantees that the end effector tracking errors are driven to zero within a fixed, pre-settable time upper bound.
Lyapunov-based stability analysis formally proves the fixed-time stability of the overall closed-loop system. Simulation studies on a realistic 6-DOF UVMS model validate the theoretical claims, demonstrating rapid, accurate, and consistent convergence from various initial conditions under significant time-varying disturbances. A quantitative performance comparison with other advanced SMC methods underscores the superior tracking accuracy and convergence speed of the proposed approach. This work provides a reliable and high-performance control framework for UVMS, enhancing the feasibility and success rate of complex underwater manipulation tasks requiring precise end effector control.